State machine and generator for generating a description of a state machine feedback function

ABSTRACT

An embodiment of a state machine for generating a pseudo-random word stream, each word of the word stream including a plurality of subsequent bits of a pseudo-random bit sequence includes a plurality of clock registers and a feedback circuit coupled to the registers and adapted to provide a plurality of feedback signals to the registers based on a feedback function and a plurality of register output signals of the registers, wherein the state machine is configured such that a first word defined by the plurality of register output signals includes a first set of subsequent bits of a pseudo-random bit stream and such that a subsequent second word defined by the plurality of register output signals includes a second set of subsequent bits of a pseudo-random bit stream.

Embodiments according to the present invention relate to state machines,generators for generating a description of a state machine feedbackfunction, methods for generating pseudo-random word stream, methods forgenerating a description of a state machine feedback function and asystem for generating a pseudo-random word stream, which may be appliedin a wide field of applications.

BACKGROUND OF THE INVENTION

Random numbers or pseudo-random numbers may be used in a wide field ofapplications in modern technology, for instance, in the field ofcryptography, numerical simulation and the generation of noise signals,to name but a few. Random numbers or pseudo-random numbers maytechnically be generated by a true random number generator or apseudo-random number generator, respectively.

While a true pseudo-random number generator typically employs physical,real source of randomness, a pseudo-random number generator typicallydoes not employ an element showing a statistical true randomness.Examples of such elements are, for instance, electric resistors withNyquist noise, electrical tunneling elements or elements based on radioactive decay with a shot noise behavior or closed feedback loopsemploying the feedback signal as the noise source to name but a few.

In contrast pseudo-random number generators typically do not employ suchelements of true randomness but are based on a deterministic behaviorsuch that the numbers generated by a pseudo-random number generator arenot true random numbers.

Thus, true random number generators and pseudo-random number generatorsare utilized whenever a random or pseudo-random value is important oradvisable to be used in a field of application. The above-mentionedexamples are a mere scratch at the surface of possible applications.

Pseudo-random number generators, compared to true random numbergenerators, typically allow a faster generation of the numbers, are veryoften more easily to implement and show a lower energy consumption thancorresponding true random number generators. However, it should be keptin mind that the pseudo-random number generators are deterministicsystems. As a consequence, the generated numbers may be calculated basedon the knowledge of the inner structure of a pseudo-random numbergenerator and the knowledge of a state of it. This may be a disadvantageunder some circumstances and a desired and wanted consequence underother circumstances, when, for instance, a periodic behavior orreproducibility may be important in some applications.

In digital implementations or applications the pseudo-random numbersgenerated by a pseudo-random number generator typically comprise digitalor binary units. Depending on the concrete implementation, apseudo-random number generator may, in such a case, provide apseudo-random bit stream (PRBS) comprising a sequence of individual bitsor a pseudo-random word stream (PRWS) comprising a sequence of words.Each word typically comprises a plurality of bits, e.g. 4 bits, 6 bits,8 bits, 16 bits, 32 bits or the like. In other words, a word of a wordstream typically comprises a plurality of bits and the word may beconsidered as a part of a sequence of pseudo-random bits. Therefore, inmany applications, a pseudo-random bit stream and a pseudo-random wordstream may be derived from each other.

Due to the ever increasing operational speeds and frequencies ofcircuits, computer systems and other devices, a demand for a fastergeneration of pseudo-random bit streams and word streams with asufficiently good statistical distribution concerning randomness of thebits exists, for instance, to allow faster processing speeds ofnumerical simulations, cryptographic applications, generation of noisesignals or other data, information and signals depending on theapplication in mind. To generate, for instance, the pseudo-random wordstream at a speed of 1 GHz, with each word comprising 4 bits entails ageneration of 4 gigabits per second (Gbps) which may represent a hightechnical challenge.

SUMMARY

According to an embodiment, a state machine for generating apseudo-random word stream, each word of the word stream including aplurality of subsequent bits of a pseudo-random bit sequence, may have:a plurality of clocked registers, the registers including correspondinginputs and corresponding outputs; and a feedback circuit coupled to theinputs and to the outputs of the registers and adapted to provide aplurality of feedback signals to the inputs of the registers based on afeedback function and the plurality of register output signals of theregisters, the register output signals being indicative of states of theregisters, wherein the state machine is configured such that a firstword defined by the plurality of register output signals includes afirst set of subsequent bits of the pseudo-random bit stream and suchthat a subsequent second word defined by the plurality of registeroutput signals includes a second set of subsequent bits of thepseudo-random bit stream; wherein the feedback circuit includes a logicstage and an adder; wherein the logic stage is configured to provide aplurality of logic stage output signals on the basis of the outputsignals of the registers by selectively weighing the output signals ofthe registers depending on a set of multi-transmission matrix elements;wherein the logic stage includes a masking gate configured to provide alogic stage output signal depending on an output signal of one of theregisters and one of the multi-transition matrix elements; wherein theadder is configured to provide at least one feedback signal using amodulo-2 summation of the logic stage output signals; and wherein theadder includes a plurality of XOR-gates or XNOR-gates for the modulo-2summation.

According to another embodiment, a method for generating a pseudo-randomword stream, each word of the word stream including a plurality ofsubsequent bits of a pseudo-random sequence, may have the steps of:outputting a first word defined by a plurality of states including afirst set of subsequent bits of the pseudo-random bit stream; alteringthe plurality of states based on a feedback function and the pluralityof states to acquire new states based on selectively weighing the statesof the registers depending on a set of multi-transmission matrixelements and based on a modulo-2 summation using a plurality ofXOR-gates or XNOR-gates for the modulo-2 summation; and outputting asubsequent second word defined by the new states including a second setof subsequent bits of the pseudo-random bit stream.

According to another embodiment, a generator for generating adescription of a state machine feedback function on the basis of a setof generator polynomial coefficients defining a configuration of alinear feedback shift register, may have: a calculator configured tocalculate the description of the state machine feedback function for astate machine on the basis of the set of generator polynomialcoefficients, wherein the calculator is configured to determine thefeedback function such that a state transition between immediatelysubsequent states of the state machine defined by the feedback functioncorresponds to a sequence of state transitions of a linear feedbackshift register, the linear feedback shift register being configured inaccordance with the generator polynomial coefficients; wherein thecalculator is configured to acquire, as an intermediate result, a singletransition matrix description of a state machine feedback function onthe basis of the generator polynomial coefficients of the linearfeedback shift register, the single transition matrix descriptiondescribing a relationship between an initial state of a state machineand a consecutive state of the state machine according to the generatorpolynomial coefficients for a single linear feedback shift registerstate transition; wherein the calculator is configured to compute apower of the single transition matrix description to acquire, as thestate machine feedback function, a multi-transition matrix descriptiondescribing, in a combined form, a plurality of linear feedback shiftregister state transitions; and wherein the calculator is adapted tocalculate the multi-transition matrix description A including themulti-transition matrix elements A_(ij) based on the equation A=T^(K)mod 2 wherein T is the single transition matrix description includingthe single transition matrix elements T_(ij), wherein i is a row indexin the range between 1 and G, wherein j is a column index in the rangebetween 1 and G, wherein N is a number of registers of the linearfeedback shift register (100), wherein R is an integer larger than N,wherein G is an integer larger than or equal to a maximum of N and R,wherein T_(1j) for j in the range between and N is equal to the set ofgenerator polynomial coefficients (310), wherein T_(ij) for i=j+1 and jin the range between 1 and (G−1) is equal to 1, and wherein T_(ij) isequal to 0.

According to another embodiment, a method for generating a descriptionof a state machine feedback function on the basis of a set of generatorpolynomial coefficients defining a configuration of a linear feedbackshift register may have the steps of: calculating the description of thestate machine feedback function for a state machine on the basis of theset of generator polynomial coefficients such that a state transitionbetween immediately subsequent states of the state machine defined bythe feedback function corresponds to a sequence of state transitions ofthe linear feedback shift register configured in accordance with agenerator polynomial coefficients wherein calculating the descriptionincludes generating a set of parameter coefficients A_(ij) such thatA=T^(R) mod 2, wherein A_(ij) are elements of the matrix A, wherein T isa matrix including elements T_(ij), wherein i is a row index in therange between 1 and G, wherein j is a column index in the range between1 and G, wherein G is an integer larger than or equal to a maximum of Nand R, wherein A_(1j) for j in the range between 1 and N is given by theset of generator polynomial coefficients (310), wherein T_(ij) for i=j+1and j in the range between 1 and (G−1) is equal to 1, wherein T_(ij) iselse equal to 0.

According to another embodiment, a state machine for generating apseudo-random word stream, each word of the word stream including aplurality of subsequent bits of a pseudo-random bit sequence, may have:a plurality of clocked registers, the registers including correspondinginputs and corresponding outputs; and a feedback circuit coupled to theinputs and to the outputs of the registers and adapted to provide aplurality of feedback signals to the inputs of the registers based on afeedback function and the plurality of register output signals of theregisters, the register output signals being indicative of states of theregisters, wherein the state machine is configured such that a firstword defined by the plurality of register output signals includes afirst set of subsequent bits of the pseudo-random bit stream and suchthat a subsequent second word defined by the plurality of registeroutput signals includes a second set of subsequent bits of thepseudo-random bit stream; wherein the feedback circuit is adapted suchthat the plurality of feedback signals provided to the inputs of theregisters are based on a single application of the feedback functionleading to new states of the registers, the new states representing astate of a linear feedback shift register initialized with the originalstates of the registers after being clocked multiple times; and whereinthe feedback function includes a set of parameter coefficients A_(ij)such that A=T^(K) mod 2, wherein A_(ij) are elements of the matrix A,wherein T is a matrix including elements T_(ij), wherein i is a rowindex in the range between 1 and G, wherein j is a column index in therange between 1 and G, wherein G is an integer larger than or equal to amaximum of N and R, wherein A_(1j) for j in the range between 1 and N isgiven by the set of generator polynomial coefficients (310), whereinT_(ij) for i=j+1 and j in the range between 1 and (G−1) is equal to 1,wherein T_(ij) is else equal to 0.

According to another embodiment, a method for generating a pseudo-randomword stream using a state machine, each word z(k) of the word streamincluding R bits, based on a set of parameter coefficients A_(ij)according to A=T^(K) mod 2, wherein A_(ij) are elements of a matrix A,wherein T is a matrix including elements T_(ij), wherein i is a rowindex in the range between 1 and G, wherein j is a column index in therange between 1 and G, wherein G is an integer larger than or equal to amaximum of N and R, wherein A_(ij) for j in the range between 1 and N isgiven by a set of generator polynomial coefficients including N bitsindicative of a feedback polynomial, wherein N is an integer larger than1, wherein R is a positive integer larger than N; wherein T_(ij) fori=j+1 and j in the range between 1 and (G−1) is equal to 1, whereinT_(ij) is else equal to 0, may have the steps of: generating the wordz(k) in a plurality of clocked registers by providing feedback signalsfrom a feedback circuit to inputs of the clocked registers such thatz(k+1)=A·z(k) mod 2, wherein z_(i)(k) is a vector element of the wordz(k) in the form of a vector, wherein k is a time index.

According to another embodiment, a state machine for generating apseudo-random word stream, each word of the word stream including aplurality of subsequent bits of a pseudo-random bit sequence, may have:a plurality of clocked registers, the registers including correspondinginputs and corresponding outputs; a feedback circuit coupled to theinputs and to the outputs of the registers and adapted to provide aplurality of feedback signals to the inputs of the registers based on afeedback function and the plurality of register output signals of theregisters, the register output signals being indicative of states of theregisters, wherein the state machine is configured such that a firstword defined by the plurality of register output signals includes afirst set of subsequent bits of the pseudo-random bit stream and suchthat a subsequent second word defined by the plurality of registeroutput signals includes a second set of subsequent bits of thepseudo-random bit stream; and a circuit to provide a word of thepseudo-random bit stream in a sequentially encoded form.

According to another embodiment, a method for generating a pseudo-randomword stream, each word of the word stream including a plurality ofsubsequent bits of a pseudo-random bit sequence may have the steps of:providing a first word defined by a plurality of states including afirst set of subsequent bits of the pseudo-random bit stream; alteringthe plurality of states based on a feedback function and the pluralityof states to acquire new states; providing a subsequent second worddefined by the new states including a second set of subsequent bits ofthe pseudo-random bit stream; and providing a word of the word stream ina sequentially encoded form.

According to another embodiment, a system for generating a pseudo-randomword stream, each word of the word stream including a plurality ofsubsequent bits of a pseudo-random bit sequence, may have: a statemachine according to any of the claims 1, 19 and 21; and a generator forgenerating a description of a state machine feedback function on thebasis of a set of generator polynomial coefficients defining aconfiguration of a linear feedback shift register, which generator mayhave: a calculator configured to calculate the description of the statemachine feedback function for a state machine on the basis of the set ofgenerator polynomial coefficients, wherein the calculator is configuredto determine the feedback function such that a state transition betweenimmediately subsequent states of the state machine defined by thefeedback function corresponds to a sequence of state transitions of alinear feedback shift register, the linear feedback shift register beingconfigured in accordance with the generator polynomial coefficients;wherein the calculator is configured to acquire, as an intermediateresult, a single transition matrix description of a state machinefeedback function on the basis of the generator polynomial coefficientsof the linear feedback shift register, the single transition matrixdescription describing a relationship between an initial state of astate machine and a consecutive state of the state machine according tothe generator polynomial coefficients for a single linear feedback shiftregister state transition; wherein the calculator is configured tocompute a power of the single transition matrix description to acquire,as the state machine feedback function, a multi-transition matrixdescription describing, in a combined form, a plurality of linearfeedback shift register state transitions; and wherein the calculator isadapted to calculate the multi-transition matrix description A includingthe multi-transition matrix elements A_(ij) based on the equationA=T^(R) mod 2 wherein T is the single transition matrix descriptionincluding the single transition matrix elements T_(ij), wherein i is arow index in the range between 1 and G, wherein j is a column index inthe range between 1 and G, wherein N is a number of registers of thelinear feedback shift register (100), wherein R is an integer largerthan N, wherein G is an integer larger than or equal to a maximum of Nand R, wherein for j in the range between 1 and N is equal to the set ofgenerator polynomial coefficients (310), wherein T_(ij) for i=j+1 and jin the range between 1 and (G−1) is equal to 1, and wherein T_(ij) isequal to 0, wherein the feedback circuit of the state machine is adaptedsuch that the feedback signals are provided based on the description ofthe state machine feedback function calculated by the generator.

Another embodiment may have a computer program for performing, whenrunning on a processor, a method according to any of the claims 11, 18,20 and 22.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will be detailed subsequentlyreferring to the appended drawings, in which:

FIG. 1 a shows an implementation of a linear feedback shift register(LFSR) with 4 bits;

FIG. 1 b shows the LFSR of FIG. 1 a in a parallel read-out mode ofoperation;

FIG. 2 shows a state machine according to an embodiment of the presentinvention;

FIG. 3 shows a generator for generating a description of a state machinefeedback function according to an embodiment of the present invention;

FIG. 4 illustrates a matrix description of the LFSR of FIG. 1 a;

FIG. 5 illustrates a matrix description for a state machine according toan embodiment of the present invention;

FIG. 6 illustrates an implementation of a state machine according to anembodiment of the present invention;

FIG. 7 illustrates a possible implementation of a feedback circuit of astate machine according to an embodiment of the present invention; and

FIG. 8 illustrates a comparison of the results of the LFSR of FIG. 1 aand a corresponding state machine according to an embodiment of thepresent invention.

DETAILED DESCRIPTION OF THE INVENTION

Before embodiments according to the present invention will be describedwith reference to FIGS. 2 to 8, a conventional pseudo-random numbergenerator in the form of a linear feedback shift register (LFSR) will bedescribed first to outline the basic principles and implementationdetails of a pseudo-random number generator.

As mentioned above, a main difference between a true pseudo-randomgenerator (TRNG) and a pseudo-random number generator (PRNG) is that thelatter typically does not involve an element offering the possibility toaccess a real physical or statistical random distribution. Apseudo-random number generator is, therefore, typically a deterministicsystem which allows a calculation of the pseudo-random bits or words andthe corresponding states of the LFSR based on the knowledge of theinternal structure of the LFSR and an initial state.

This lack of true randomness may have under some circumstances anegative effect and under some circumstances a positive effect, when,for instance, a periodic behavior or a reproducibility is wanted ordesired. For instance, it may be interesting to generate a deterministicbit stream comprising a known number of words with a known number ofones and zeroes. For instance, it may be favorable to generate aperiodic pseudo-random word stream comprising a known number of wordshaving (N−1) ones and a known number of words having (N−1) zeroes, whereN is the length of each of the words.

It is also often accompanied by the possibility of a faster and lessenergy consuming generation of (pseudo-) random numbers compared to atrue pseudo-random number generator. Furthermore, a pseudo-random numbergenerator is in many cases easier to implement since an implementationis mainly based on arithmetic or logical calculations compared to animplementation of an element offering access to an effect with a truerandom statistic. Not only due to these possible advantages ofpseudo-random number generators, they are widely employed in many fieldsof applications where a (pseudo-) random number may be generated.

A pseudo-random number generator may conventionally be implemented as alinear feedback shift register (LFSR), which will be described in moredetail for a (LFSR) with N=4 registers in the context of FIG. 1 a. Alinear feedback shift register comprises a cascade of simultaneously orparallel clock registers of which the first is provided with a feedbacksignal based on one or more states of the registers. When more than onestate of the registers is used for the generation of the feedback signalfor the first register typically a XOR logic gate (XOR=exclusive OR) isused. Since all these electrical elements can easily be implemented inhardware, linear feedback shift registers may easily be implemented andrealized in hardware.

Depending on the feedback signal provided to the first register, linearfeedback shift registers are capable of generating a pseudo-random bitstream or pseudo-random word stream with statistical properties wellenough even for cryptographic applications. This is for instance true,when the feedback signal for the first register is based on a feedbackfunction or tap sequence which corresponds to a primitive polynomial.This, however, will be described in more detail in context with FIGS. 4,5 and 8.

FIG. 1 shows an example of a LFSR 100 with N registers 110. To be moreprecise, the LFSR 100 of FIG. 1 a comprises N=4 registers 110-1, 110-2,110-3, 110-4. Each of the registers comprises an input 110 a, an output110 b and a clock signal input 110 c, which are only labeled for thefirst register 110-1 for the sake of simplicity and clearness of thepresentation of FIG. 1 a.

The four registers 110-1, 110-2, 110-3 and 110-4 are connected to form adaisy chain such that, apart from the last 110-4, the outputs 110 b ofthe registers 110 are coupled to the inputs 110 a of the immediatelyfollowing or consecutive register 110. In other words, the output 110 bof the first register 110-1 is coupled to the input 110 a (not labeledin FIG. 1 a) of the second register 110-2. Accordingly, also the outputsof the second and third register 110-2, 110-3 are coupled to the inputsof the third and fourth register 110-3, 110-4 respectively.

The input 110 a of the first register 110-1 is coupled to an output of aXOR gate 120, which comprises two inputs coupled to the output 110 b ofthe first register 110-1 and to the output 110 b of the fourth register110-4, respectively. The XOR gate 120 has generated the feedback signalprovided to the input 110 a of the first register 110-1 based on anexclusive or combination of the states of the first and the fourthregisters 110-1, 110-4.

An output of the linear feedback shift register 100 represents theoutput 110 b of the fourth register 110-4 at which the current statex₄=x_(N) is generated as the pseudo-random bit y.

In FIG. 1 a the connection of the clock signal inputs 110 c of the fourregisters 110 have been omitted for the sake of clarity. The clocksignal inputs 110 may be connected in parallel to an input for the clocksignal of the linear feedback shift register 100.

Since the registers 110 of the linear feedback shift register 100 shownin FIG. 1 a are adapted to store two states and are adapted to acquire astate in response to a signal present at the respective inputs 110 a andan appropriate clock signal present at the clock signal 110 c, one bitof a pseudo-random bit stream is generated for each clock cycle of aclock signal presented to the linear feedback shift register 100. Sinceat their respective outputs 110 b a signal indicative of the presentstate of the register 110 is available, a pseudo-random bit stream orpseudo-random bit sequence {y(0), y(1), . . . , y(L)} will be present atthe output of the linear feedback shift register 100.

The linear feedback shift register 100 shown in FIG. 1 a is described bygenerating characteristic polynomial g(x), which defines the tabsequence of feedback tabs. The characteristic polynomial g(x) of thefeedback function for the linear feedback shift register 100 of FIG. 1 ais given byg(x)=1+x+x ⁴,  (1)which indicates that the output of the first register 110-1 (summandx=x¹) and the output of the fourth register 110-4 (summand x⁴) are used.

In other words, an equivalence between the characteristic polynomialg(x) and an implementation of a linear feedback shift register 100exists, which can be described by summing the outputs of all registerswith the corresponding number represented as the power of the summand ofx. Typically, an additional summand 1 is also added indicating thefeedback.

The LFSR 100 shown in FIG. 1 a comprises N=4 registers 110 so that itcan acquire 2^(N)=2⁴=16 different states 0000, 0001, . . . , 1111. Sincethe LFSR 100 initialized with a state 0000 leads to a feedback signalprovided to the input 110 a of the first register 110-1, which is equalto 0, no change of the states of the registers of the LFSR 100 willoccur. Hence, the state 0000 will not lead to any change of the statesof the LFSR 100. The maximum sequence length of a pseudo-random bitstream generated by the LFSR 100 is 2^(N)−1 which will be acquired onlyfor so-called primitive characteristic polynomials. Embodimentsaccording to the present invention, are neither limited to primitivepolynomials nor to non-primitive polynomials, as will be outlined inmore detail below.

Due to the daisy chain configuration or series connection of theregisters 110, the next N−1 bits of the pseudo-random bit stream are thecurrent states x_(N−1)(k), . . . , x₁(k) of the registers 110-3, . . . ,110-1.

The linear feedback shift register 100 shown in FIG. 1 a represents astraightforward implementation of a pseudo-random bit stream generation.However, the linear feedback shift register 100 as shown in FIG. 1 a canalso be used to generate a pseudo-random word stream. While apseudo-random bit stream comprises a sequence of single bits, apseudo-random word stream comprises a sequence of words of, forinstance, 4 bits, 6 bits, 8 bits, 16 bits, 32 bits or the like. Hence, aword comprises a plurality of bits or a part of a sequence ofpseudo-random bits. This essentially means that a pseudo-random bitstream and a pseudo-random word stream may be, under some circumstances,transferred into one another. An example for such a transferal isdescribed in context of FIG. 1 b.

FIG. 1 b shows, the linear feedback shift register 100 shown in FIG. 1 ain a configuration for generating a pseudo-random word stream instead ofa pseudo-random bit stream. To achieve this, the linear feedback shiftregister 100 of FIG. 1 b differs from that of FIG. 1 a only with respectto the configuration of the output of the linear feedback shift register100. While in the case of the LFSR 100 of FIG. 1 a the output of theLFSR mainly represents the output 110 b of the fourth register 110-4,the output of the LFSR 100 of FIG. 1 b is coupled to all outputs of thefour registers 110-1, . . . , 110-4. The output of the LFSR 100 providesas bit y (k) the signal present at the output of the register 110-4, asbit k+1 the signal present at the output of third register 110-3, as bitk+2 the signal present at the output of the second register 110-2 and asbit k+N−1=k+3 the output of the first register 110-1.

The LFSR 100 of FIG. 1 b produces the same pseudo-random bit sequence orpseudo-random bit stream as the LFSR 100 of FIG. 1 a by the previouslydescribed parallel read-out the N=4 registers and afterwards clockingthe LFSR N=4 times, while the output during these clocking sequences isignored. Naturally, the LFSR 100 as shown in FIG. 1 b cannot only beused to obtain pseudo-random words with N bits each but any number ofbits R being smaller than or equal to the number of registers N of theLFSR 100. In this case, the parallel read-out may be limited to R stagesand a subsequent clocking of the LFSR 100 R times, while ignoring theoutput during these intermediate clockings.

Naturally, as indicated earlier, different feedback functions than theone as defined by the characteristic polynomial g(x) occurring toequation (1) may be implemented. Also, longer or shorter linear feedbackshift registers 100 with a different number of registers N may beimplemented. Also the clock signal provided to the clock signal input100 c of the registers 110 may optionally be generated inside the LFSR100 instead of being provided externally.

The pseudo-random number generators 100 in the form of the linearfeedback shirt registers 100 as depicted in FIGS. 1 a and 1 b are,however, comparably slow in the case of generating pseudo-random wordsfor a pseudo-random word stream, wherein each word comprises more thanone bit. In these cases the linear feedback shift register 100 may beclocked multiple times in the case of a sequential readout of apseudo-random bit stream as illustrated in FIG. 1 a as well as in thecase of a parallel readout as indicated in FIG. 1 b. In other words, thegeneration of pseudo-random words for words comprising R bits (R≦N) mayuse R clock cycles of a clock signal provided to the registers 110. Theclock signal provided to the LFSR 100 may have a frequency of R-timesthat of a desired pseudo-random word stream.

However, apart from the previously described drawback of the comparablyslow generation of pseudo-random words, a linear feedback shift registeroffers a statistically well distributed pseudo-random bit stream,especially when operated based on a primitive polynomial. Hence, ademand exists to generate a pseudo-random word stream with a comparablelevel of quality concerning the statistical distribution ofpseudo-random bits of a LFSR faster.

FIG. 2 shows a block diagram of a state machine 200 according to anembodiment of the present invention. The state machine 200 comprises aplurality of clocked registers 210, each register 210 comprising aninput 210 a for a feedback signal or an input signal and an output 210 bfor an output signal. Moreover, the registers 210 further comprise aclock signal input 210 c for a clock signal. Similar to the registers110 of FIGS. 1 a and 1 b, also the registers 210 are adapted to store atleast two states in response to a signal presented to their respectiveinputs 210 a and a corresponding clock signal presented to the clocksignal input 210 c. The registers 210 are adapted to provide at theiroutputs 210 b a signal indicative of their state.

The block diagram of FIG. 2 shows a first register 210-1 and a secondregister 210-2, which are coupled to a common clock signal line 220 toprovide the registers 210 with a corresponding clock signal in parallelvia their clock signal inputs 210 c. Moreover, the state machine 200 asshown in FIG. 2 comprises a feedback circuit 230 with a plurality ofinputs 230 a and a plurality of outputs 230 b. The number of inputs 230a corresponds at least to the number N of registers 210 of the statemachine 200 so that each of the registers 210 is connected to anindividual input of the feedback circuit 230 in the embodiment shownhere. Accordingly, also the number of outputs 230 b of the feedbackcircuit 200 corresponds at least to the number of registers 210 so thateach of the registers 210 is connected via its input 210 a to one of theoutputs 230 b of the feedback circuit 230. However, in other embodimentsaccording to the present invention the number of inputs or outputs ofthe feedback circuit 230 may be smaller or larger.

The plurality of registers 210 and the feedback circuit 230 form aclosed feedback loop, wherein the feedback circuit 230 generates aplurality of feedback signals which are provided to the inputs 210 a ofthe registers 210. Since the registers 210 are adapted to assume a statecorresponding to a signal provided to the input 210 a upon reception ofa clock signal (e.g. an edge or a transition or level of the clocksignal) provided via the clock signal line 220, the feedback circuit 230influences the change of states via a creation of the feedback signalsfrom clock cycle to clock cycle of the clock signal. Sometimes aregister is also referred to as a flip-flop. In other words, a register,such as the registers 210, are adapted to store at least two differentstates. The state to be stored is provided to the input 210 a of theregister 210 and will be stored therein upon reception of a clocksignal. In contrast, a latch may be transparent or freeze a statedepending on a control signal provided to a control signal input.

However, before describing the functionality of the state machine 200 asshown in FIG. 2 in more detail, it should be noted that the outputs 210b of the registers 210 are coupled directly or indirectly to an output240 of the LFSR 200 so that not only the output signals of the registers210 are provided to the feedback circuit 230 but also to the output 240.

The feedback circuit 230 is output such that depending on a feedbackfunction and the signals provided to the inputs 230 a of the feedbackcircuit 230 the feedback signals are generated and provided via theoutputs 230 b to the inputs 210 a of the registers 210. The feedbackcircuit 230 is, in this context, adapted such that a single applicationof the feedback function leads to new states of the registers 210representing a state of a linear feedback shift register, for instance,as shown in FIG. 1 a or 1 b, initialized with the original states of theregisters 210 after being clocked multiple times. In other words, thefeedback function as implemented in the feedback circuit 230 offers thepossibility of generating feedback signals and providing them to theinputs 210 a of the registers 210 which correspond to a multipleapplication of clock cycles of a clock signal to a LFSR initialized withstates as present in the registers 210 of the state machine 200 beforeproviding the appropriate clock signal to the clock signal line 220.

A single application of the feedback function as implemented in thefeedback circuit 230 results in the generation of feedback signals suchthat immediately consecutively generated pseudo-random words at theoutputs 210 b of the registers 210 in response to the clock signalrepresent the change of states which in the case of a LFSR may usemultiple clock cycles. In embodiments according to the presentinvention, this may lead to the feedback circuit 230 generating at leasttwo feedback signals based on register output signals of at least twodifferent registers 210, as will be explained in the context of FIGS. 4,5 and 8 in more detail. However, it is important to note that inembodiments according to the present invention, the feedback signalsonly depend on the feedback function and the plurality of registeroutput signals. Further parameters are typically not required, althoughthe feedback function itself may be programmable or provided to afeedback circuit 230 which will be described in more detail later on.

Hence, embodiments according to the present invention are based on thefindings that a plurality of bits of a pseudo-random bit stream may begenerated in parallel to form pseudo-random words of a pseudo-randomword stream by implementing a state machine 200 with a plurality ofparallel connected registers 210 to a feedback circuit 230 generating aplurality of feedback signals and providing same to the inputs 210 a ofthe registers 210. Embodiments according to the present invention in theform of a state machine 200, therefore, enable a faster generation of apseudo-random word stream since, for each clock cycle not just onefeedback signal, as in the case of a LFSR, is generated, but a pluralityof feedback signals, each provided to one of the registers 210 withoutsacrificing the statistical quality of the generated pseudo-random bitstream of a LFSR.

A state machine 200 according to an embodiment of the present inventionmay offer, therefore, the possibility of generating an almost arbitrarypseudo-random bit stream for at-speed tests, which may be an interestingmotivation for implementing such. Moreover, embodiments according to thepresent invention may be implemented in a field of interest in which adirect implementation of a linear feedback shift register and at fullrate of, for instance 8 or 10 Gbps or more is an extremely challengingtask. Depending on the actual frequency of the clock signal itself, ageneration of the pseudo-random words is limited in terms of speed,mostly by the speed of the registers 210, the signal propagation time ofthe interconnecting signal lines and propagation delay of the feedbackcircuit 230. For instance, the propagation delay of the feedback circuit200 should be lower than the period of a single period of the clocksignal. Hence, based on a clock cycle frequency of 1 GHz, thepropagation delay of the feedback circuit 230 should not be longer than1 ns. To put it in more general terms, the propagation delay of afeedback circuit 230 should be smaller than 1/f wherein f is thefrequency of the clock signal as provided to the registers 210.

The state machine 200 as shown in FIG. 2 may optionally comprise acircuit 250 coupled in between the outputs 210 b of the registers 210and the output 240 of the state machine 200. The circuit 250 may forinstance be a circuit adapted to generate from the parallel providedoutput signals of the registers 210 a sequentially encoded pseudo-randomword and provide same to the output 240 of the state machine 200.Moreover, the circuit 250 may also comprise further components orcircuits on the basis of which the pseudo-random bits as provided by theregisters 210 in parallel to the circuit 250 are transferred into anappropriate output signal of the state machine 200. For instance, in thecase of a noise generator, the circuit 250 may comprise adigital/analogue converter, a look-up table or another component totransfer the parallel digital information encoded in the signals of theregisters 210 into a desired output signal provided the output 240 ofthe state machine 200.

Moreover, as indicated by a dashed arrow 260 the feedback circuit 230may be programmable by providing a programming signal to an input 260 ofthe feedback circuit 230 so that the feedback circuit may depend on oneor more parameters as provided to the feedback circuit 230. This,however, is an optional component which is by far not required in thecase of an implementation of the feedback circuit 230 according to anembodiment of the present invention. This, however, offers theopportunity to implement a generation of a pseudo-random bit streampolynomial being programmable. As will be outlined in more detail in thecontext of FIGS. 6 and 7, this may, for instance, be achieved on thebasis of a modular-2 scalar product of a set of multi-transmissionmatrix elements and the values of the register output signals asprovided to the inputs 230 a of the feedback circuit 230. This will bedescribed in more detail with respect to both, the mathematicalbackground and an implementation in more detail below.

A way to determine the previously mentioned multi-transmission matrixelements which enables a generation of multiple bits in one cycle of theclock signal for programmable polynomials will be described in moredetail below. Hence, FIG. 2 shows an apparatus containing a programmablestate machine that generates multiple bits of a pseudo-random bit streamwith optionally arbitrary polynomials in one clock cycle. Apart from thementioned multi-transmission matrix elements, also other programmingsignals may be provided to the input 260 of the feedback circuit 230. Inother embodiments according to the present invention, the programmingsignal may comprise a truth table, a script or a program code in alanguage for programming a LPGA or the like.

However, before describing a more concrete implementation of a statemachine 200 according to an embodiment of the present invention, withrespect to FIGS. 3, 4 and 5, a generator for generating a description ofa state machine feedback function and the basic principle behind agenerator according to an embodiment of the present invention will bedescribed first.

FIG. 3 shows a schematic block diagram of a generator 300 for generatinga description of a state machine feedback function on the basis of a setof generator polynomial coefficients 310. The generator 300 comprises asa central component a calculator 320 to which the generator polynomialcoefficients 310 are provided. The calculator 320 is configured todetermine a state transmission between immediately subsequent states ofthe state machine defined by the feedback function corresponding to asequence of state transitions of a linear feedback shift register. Theoperation of a linear feedback shift register is, in this context, basedon the polynomial corresponding the generator polynomial coefficients310.

In some embodiments according to the present invention, the calculator320 is adapted to optionally generate a single transition matrix 330 aswill be described in more detail in the context of FIGS. 4 and 5, andafterwards, based on the single transition matrix 330, amulti-transition matrix 340. The multi-transition matrix elements of thematrix 340 may then be provided as the description of the feedbackfunction at an output 350 of the generator 300, for instance, to theoptional input 260 of the feedback circuit 230 shown in FIG. 2.

However, it should be noted that a generator 300 according to anembodiment of the present invention is by far not required to beingbased on the calculation of the single transition matrix 330 and themulti-transition matrix 340. Embodiments according to the presentinvention utilizing a different technique will be laid out in moredetail below. However, in the following first of all the generator 300being based on the calculation of the two previously mentioned matrices330, 340 will be described.

A first observation on which an implementation of the generator 300using the two matrices 330, 340 is based on, is that a linear statemachine such as the state machine 200 according to an embodiment of thepresent invention, but also the linear feedback shift register as, forinstance, shown in FIGS. 1 a and 1 b, can be described by a statetransition matrix T. In the case of a digital or binary implementationof a generator 300 according to an embodiment of the present invention,for being used in context with a digital or binary implementation of astate machine 200 according to an embodiment of the present invention,it should be noted that the corresponding mathematical relations aredefined over the corresponding mathematical field, which is in this casethe Galois field GF(2). Accordingly, adding or summing up elements inthe field corresponds to a XOR-combination of the correspondingelements. The other combination in the field, the multiplication,corresponds to an AND-combination of the respective elements to bemultiplied with each other. Calculations in this field are sometimesalso referred to as modular-2 calculations.

Hence, the state machine 200 according to an embodiment of the presentinvention as well as a linear feedback shift register as, for instance,shown in FIGS. 1 a and 1 b, may be, described by a state-transitionmatrix T with XOR-combination as an addition or formation and aAND-combination as a multiplication.

To illustration this, FIG. 4 shows at the bottom the linear feedbackshift register 100 of FIGS. 1 a and 1 b. As described in the context ofequation (1), the LFSR 100 shown in the lower part of FIG. 4 is based onthe generator polynomial 400, which is extended by a summandscorresponding to the vanishing contributions compared to equation (1).Defining the states of the four registers 110-1, . . . , 110-4 as x₁(k),. . . , x₄(k), wherein k is an index indicating a number of statetransitions, which may in some embodiments according the presentinvention be equal to or depend on the number of clock cycles of theclock signal.

With a vector x(k) given by

$\begin{matrix}{{x(k)} = \begin{pmatrix}{x_{1}(k)} \\\vdots \\{x_{N}(k)}\end{pmatrix}} & (2)\end{matrix}$for a linear feedback shift register 100 with N registers 110, thetransition to the next state is given byx(k+1)=T·x(k)  (3)with a state-transition matrix T. Since the state-transition matrix Trepresents a single transition of the states x₁, . . . , x_(N), it isalso referred to as a single transition matrix or single transitionmatrix description.

In the case of a linear feedback shift register 100 shown in FIGS. 1 a,1 b and the bottom of FIG. 4 with N=4, the single transition matrix T isgiven by the matrix 410 as shown in FIG. 4. The first row of the matrix410 is given by a set of generator polynomial coefficients of thepolynomial 400 apart from the constant summoned 1 indicative of theXOR-gate 120 of the linear feedback shift register 100. A sub-matrix 420extending from the second row to the last row describes a shift of thestates of the registers 110 in response to a clock signal which iscaused by the daisy-chain configuration of the registers 110. For thesake of completeness, it should be noted that the matrix 410 as allsingle transition matrices are quadratic matrices since thecorresponding matrices describe a transition of states of registers 110so that the number of initial states and the number of states after thetransition are equal and given by the number N of registers 110 in theLFSR 100 or the number of registers 210 of the state machine 200according to an embodiment of the present invention, as will be outlinedbelow.

With the matrix 410 as the single transition matrix T, equation (3)becomes the matrix equation 430 as shown in FIG. 4. Moreover, for thesake of completeness, it should be noted that the output of the linearfeedback shift register 100 shown in FIG. 4 corresponds to the state x₄(k) of the register 110-4. Naturally, in the case of a linear feedbackshift register 100 with N registers 110 the output of a correspondingLFSR 100 is the state x_(N) of register 110-N.

Providing a linear feedback shift register as the LFSR 100 shown in thebottom of FIG. 4 with more than one clock cycle so that more than onestate transition takes place, the state vector x(k) changes according toconsecutive application of the matrix equation (3). Hence, after R clockcycles, wherein R is an integer larger than 1, the states of theregisters 110 as given by the state vector x(k+R) are given byx(k+R)=T ^(R) ·x(k),  (4)wherein equation (4) also holds true for R=1. Hence, for the applicationof equation (4) R may be any non-negative integer. Just as a side note,in equation (4) R may even be a negative integer to reestablish aprevious state, if matrix T is invertible. This illustrates thedeterministic nature of a pseudo-random number generator such as thestate machine 200 according to an embodiment of the present invention.

However, based on equation (4) the concept of a possible implementationof a generator 300 according to an embodiment of the present inventioncan be explained. Based on equation (4) it is possible to synthesize astate machine 200 according to an embodiment of the present inventionthat outputs R bits per clock cycle. The linear finite state machine 200with a state transition matrix or multi-transition matrix A based on theequationA=T ^(R)  (5)comprises R subsequent pseudo-random bits of a pseudo-random bit streamin its first R state variables corresponding to R registers 110,wherein, for the moment, R is assumed to be equal to or smaller than N,the number of registers of the underlying LFSR. The state machine 200according to an embodiment of the present invention may then proceed tothe next R bits of a pseudo-random bit stream or a next word of thepseudo-random word stream in a single clock cycle based on the equationz(k+1)=A·z(k),  (6)wherein z(k) is a state vector comprising the states z₁(k), . . . ,z_(N)(k) of the registers of 210 of the state machine 200 according toan embodiment of the present invention in accordance with

$\begin{matrix}{{z(k)} = {\begin{pmatrix}{z_{1}(k)} \\\vdots \\{z_{N}(k)}\end{pmatrix}.}} & (7)\end{matrix}$

In other words, the preceding description of the functionality of agenerator 300 according to an embodiment of the present inventionillustrates a method according to an embodiment of the present inventionfor generating a state machine description in the form of themulti-transition matrix elements for a state machine 200 according to anembodiment of the present invention that generates multiple bits of apseudo-random bit stream per clock cycle, given the correspondingpolynomial.

In the case of a state machine 200 according to an embodiment of thepresent invention which is intended to provide more bits in one clockcycle than the number of registers N of the underlying linear feedbackshift register, the number of registers 210 of the state machine 200 isextended so that a sufficient number of registers 210 exists to storethe corresponding states. In other words, in the case that the number ofbits R to be generated by the state machine 200 according to the presentinvention, corresponding to R clock cycles of the underlying linearfeedback shift register, is larger than the number N of registers 110 ofthe corresponding LFSR 100, the original shift register is extended sothat R registers of flip-flops are capable of comprising R subsequentbits of the pseudo-random bit stream. Naturally, the extension is notlimited to the number of bits to be provided (R). Hence, the statemachine 200 according to an embodiment of the present invention maycomprise G registers 210, wherein G is equal to or larger than a maximumof the number of registers N of the underlying linear feedback shiftregister and the number R of bits to be generated in one clock cycle.Accordingly, also the size of the single transition matrix T and that ofthe multi-transition matrix A according to equation (5) may comprise Gelements each direction. Both matrices are, therefore, quadraticmatrices comprising G×G elements.

To illustrate this further, FIG. 5 shows a schematic representation ofthe single transition matrix T for the case of R being larger than thenumber of registers N of the underlying LFSR 100. The single transitionmatrix T 330 comprises in a first row the elements 310 corresponding tothe generator polynomial 400. Based on the set of generator polynomialcoefficients {q_(N), q_(N−1), . . . , q₁}, corresponding to thepolynomial

$\begin{matrix}{{g(x)} = {{\sum\limits_{i = 1}^{N}\;{q_{j}x^{j}}} + 1.}} & (8)\end{matrix}$the polynomial coefficients q_(n) are entered as matrix elements T_(1n)for j=1, . . . , N. The rest of row 1, if present at all, will be filledwith a column vector 440 (0) comprising zeros. Hence, the matrix elementT_(1j) are set to 0 for j=N+1, . . . , G.

The rest of the single transition matrix T 330 is filled with a shiftmatrix S 450 being a sub-matrix of the single transition matrix T. Theshift, matrix S 450 is a (G−1)×G matrix comprising the matrix elementsS _(ij)=δ_(ij)  (9)for i=1, . . . , G and j=1, . . . , G−1, wherein δ_(ij) is theKronecker-symbol. Since the shift matrix S 450 is a sub-matrix of thesingle transition matrix T 330 starting with the second row, the singletransition matrix T comprises a diagonally extending line of ones,T_(ij)=1 for i=j+1. Apart from this line of ones and the generatorpolynomial coefficients 400, the rest of the single transition matrix T330 only comprises zeros.

In other words, the only non-vanishing elements of the single transitionmatrix T 330 may be the elements of the aforementioned diagonallyextending lines of ones of the shift matrix S 450 and optionally thosenon-vanishing coefficients of the generator polynomial coefficient 400.The term “non-vanishing” means in this context being not equal to zero,while the term “vanishing” may synonymously be used for being equal tozero, optionally within a reasonable limit defined by a precision of theimplementation.

In the case of digital or binary implementations as described in thecontext of FIGS. 2 and 3 so far, all matrix elements and polynomialcoefficients are either 0 or 1. In this case a vanishing element orcoefficient is equal to 0 and a non-vanishing element or coefficient isequal to 1.

It should be noted that the equations and formulas given in the presentdescription serve to illustrate and to describe the operationalprinciples of embodiments according to the present invention. Whenimplementing a specific embodiment, indices of the different registersmay naturally be altered. This may lead to a “rearrangement” of matricesand other equations. In other words, expressions and equations maydiffer and comprise permutations with respect to indices and otherdeviations from the formulas given here.

The calculator 320 of the generator 300 according to an embodiment ofthe present invention obtains, as an intermediary result, the singletransition matrix T 330 based on the generator polynomial coefficients310 provided to the generator 300. Then the calculator 320 generates themulti-transition matrix A based on equation (5) as laid out in FIG. 5.

Based on the multi-transition matrix description as a description of thefeedback function of the feedback circuit 230 of the state machine 200,the feedback circuit 230 is then adapted such that the feedback functionrepresents a plurality of consecutive LFSR-cycles based on the states ofthe registers 210 to yield the plurality of feedback signals indicativeof the next states of the registers 210 of the state machine 200corresponding to the states of the underlying LFSR after a correspondingnumber of clock cycles.

However, in many of the embodiments according to the present invention,the generator of the calculator 320 is adapted such that the set ofgenerator polynomial coefficients 310 comprises at least twonon-vanishing generator polynomial coefficients. Examples of this havealready been shown in equation (1) and the corresponding FIG. 4.Starting from a primitive polynomial for a linear feedback shiftregister 100, the generator polynomial coefficients 310 in many casescomprise two or more non-vanishing coefficients. Therefore, after anapplication of equation (5) the multi-transition matrix A 340 (cf. FIG.3) typically comprises for R being larger than 1 for more than one rowof the multi-transition matrix A 340 rows with two non-vanishingelements.

FIG. 6 shows a further state machine 200 according to an embodiment ofthe present invention in the form of a programmable state machineallowing a generation of a pseudo-random bit stream based on aprogrammable polynomial. The state machine 200 of FIG. 6 comprises 32registers 210-1, . . . , 210-32 which are coupled to the feedbackcircuit 230 with both, there inputs and outputs. The outputs of theregisters 210 are coupled to a 32-bit wide bus, which is coupled to aninput of the feedback circuit 230.

The feedback circuit 230 comprises a plurality of WX-gates 500-1, . . ., 500-32 for each of the registers 210-1, . . . , 210-32. Each of theWX-gates 500 comprises two 32-bit wide inputs of which one is coupled toall outputs of the registers 210-1, . . . , 210-32 and of which theother is coupled to the input 260 of the feedback circuit 230 to whichthe feedback function description in the form of the multi-transitionmatrix A 340 is provided to.

The WX gates 500 are adapted to calculate a modular-2 scalar product ofthe two 32-element wide vectors provided to their respective inputs andto generate one feedback signal and an output of the WX-gates 500, whichis coupled to the input of one of the registers 210. However, beforedescribing a possible implementation of a WX-gate 500, which is alsoreferred to as a weighted XOR sum or weighted XOR adder, as well as themulti-transition matrix elements provided to the WX-gate 500, theoutputs of the registers 210 are coupled to an output 240 of the statemachine via a circuit 250. The circuit 250 comprises 8×4:1-multiplexerunit 510, which is coupled to the 32 outputs of the registers 210. Eightsignal lines connect the multiplexer unit 510 and a parallel/serialtransformer 520, which is in turn coupled to the output 240.

From the 32 latched output signals provided by the registers 210 themultiplexer unit 510 generates four 8-bit wide output signals andprovides each of these 8-bit wide output signals to the parallel/serialtransformer 520. The parallel/serial transformer 520 comprises 8 inputsignal lines and generates at its output a serial signal comprising the8 bits received as a serial stream. In other words, the8×4:1-multiplexer unit 510 and the parallel/serial transformer 520forming the circuit 250 transform the 32 bits provided in parallel intoa serial (1 bit wide) bit stream. By reducing the width of the wordssuccessively from 32 bits to 8 bits into 1 bit, the frequency of theprovision of bits is increased accordingly so that at the output of thestate machine the bits of the PRBS are provided at a rate being 32-timeshigher than the frequency of the clock signal of the registers 210.

In other words, the 8×4:1-multiplexer unit 510 will typically be providewith a clock signal having a frequency of four times that of the clocksignal provided to the registers 210 to switch each of the eight outputlines of the multiplexer unit to one of the four connected input linescoupled to the registers 210. The multiplexer unit 510, therefore,allows in the embodiment shown in FIG. 6 the provision of four 8-bitwide pseudo-random words to the parallel/serial transformer 520 based onthe 32 states stored in the registers 210. In turn, the parallel/serialtransformer 520 will provide at its output the bits with a frequencybeing 8-times higher than the frequency of the 8×4:1 multiplexer unit510. As a consequence, the circuit 250 comprising the 8×4:1 multiplexerunit 510 and the parallel/serial transformer 520 allows a generation ofan output signal to the output 240 with a frequency 32-times higher thanthe clock signal provided to the registers 210 in the embodiment shownhere.

It should be noted that the circuit 250 as well as its components shownin FIG. 6 are optional components, which may not be implemented at allor modified. Hence, in other state machines 200 according to embodimentsof the present invention, circuits 250 allowing a generation of adifferent number of bits per clock cycle of the clock signal provided tothe registers 210 may be implemented. In other words, the frequency ofbits provided by the state machine may be any integer-multiple of thefrequency of the clock signal provided to the registers 210.

As a further optional component the circuit 250 may also comprise aFIFO-circuit (FIFO=first in-first out) which may be, for instance, usedto buffer the generated word stream. This may be for instanceinteresting in the case, where the state machine 200 cannot be operatedin a continuous mode or where the state machine 200 may be halted for areprogramming of the feedback function, but the word stream may have tobe uninterrupted. In the case of implementing a FIFO-circuit or asimilar circuit, it may be possible to operate the state machine 200 ata frequency for generating the bits of the bit stream is not necessarilyin integer multiple of the frequency of the clock signal provided to theclock signal inputs of the registers 210. In the case of animplementation with a FIFO-circuit or a similar circuit, the frequencyof the generated bits may be higher than the clock signal frequency by afactor being larger than 1. Naturally the factor may be also in thiscase an integer.

Switching back to the feedback circuit 230 and the WX-gates 500, ofwhich an implementation will be described in more detail in the contextof FIG. 7, it should be noted that a feedback function description isprovided to the input 260 in the form of a multi-transition matrix A 340as illustrated in FIG. 6. The multi-transition matrix A 340 may beconsidered as a column vector 530 comprising a plurality of row vectorsa₁, . . . , a₃₂ as vector elements, which correspond to the matrixelements A_(ij) of one row each. As already described, the WX-gates 500are adapted to calculate a modulo-2 scalar product of two 32 elementscomprising vectors provided to the two inputs of each of the WX-gates500. Based on the row vectors a_(i) for i=1, . . . , 32, each of theWX-gates 500 generates a feedback signal provided to one of theregisters 210 indicative of the next state to be acquired bycorresponding register 210 according toz _(i)(k+1)=a _(i) ·z(k).  (10)

In other words, the WX-gates 500 each generate one feedback signal basedon a 32-element comprising vector a_(i) and the 32 states z₁, . . . ,z₃₂ of the registers 210.

Since each of the row vectors a_(i) of the column vector 530 comprisesthe matrix elements of the multi-transition matrix A 340 of therespective row, the element with the index j of the row vector a_(i) isgiven by(a _(i))_(j) =A _(ij)  (11)

As a consequence, the scalar product of vector a_(j) and the vector z(k)yields the feedback signal or the subsequent state z_(i)(k+1), which isgiven by

$\begin{matrix}{{z_{i}( {k + 1} )} = {{\sum\limits_{j = 1}^{32}\;{( a_{i} )_{j} \cdot {z_{j}(k)}}} = {\sum\limits_{j = 1}^{32}{A_{ij} \cdot {{z_{j}(k)}.}}}}} & (12)\end{matrix}$

In this context, it should be remembered that in the case of a digitalor binary implementation as described before, the mathematicalcalculations of the preceding equations are to be carried out in themathematical field GF(2) so that a summation is given by aXOR-combination of corresponding elements and a multiplication isequivalent to an AND-combination.

FIG. 7 shows a part of a block diagram of the WX-gate 500 as it isimplemented in the state machine according to FIG. 6. As illustrated inFIG. 6, the state machine 200 comprises 32 instances of the WX-gate 500.Each of the WX-gates 500 comprises a logic stage 530 and an adder 540.The logic stage 530 comprises 32 AND-gates 550-1, . . . , 550-32connected in parallel, which each one of the output signals of theregisters 210 indicative of the respective states z_(j) and one elementof the multi-transition matrix A is provide to. As outlined in thecontext of equation (10), the WX-gate 500 for register 210 correspondingto state z_(i) is provide with the row vector a_(i) so that to each ofthe AND gates 550 of the logic stage 530 one of the vector elements orone of the corresponding multi-transition matrix elements according toequation (11) is provided.

In other words, each of the register output signals indicative of therespective state of the register 210 is weighted according to oneelement of the multi-transition matrix A such that the correspondingsignal is either set to a predetermined value (e.g. 0) or a remainsunchanged. Each of the AND-gates 550 represents a masking gate by whichthe corresponding register output signal may be weighted. Accordingly,also other masking gates such as OR-gate, NOR-gate or a NAND-gate may beused.

Since each of the AND-gates 550 comprises two inputs and one output, thenumber of signals output by the logic stage 530 is reduced by a factorof 2. Starting from the 64 binary signals the first 32 AND-gates 550 ofthe logic stage 530 reduce the number of signals provided by the logicstate 530 to 32.

These 32 logic state output signals are then provided to the adder 540,which comprises a cascade of 31 XOR-gates 560 with two inputs and oneoutput each. The 31 XOR-gates 560 are cascaded to reduce the number ofsignals in each layer of the cascade approximately by a factor of 2.Hence, the 32 output signals of the logic stage 530 are provided to thefirst 16 XOR-gates 560 producing 16 intermediate signals of a firstlayer. These 16 signals are then provided to another 8 XOR-gates 560which reduce the overall number once again by a factor of 2. Thefollowing layers then reduce the number of signals further, until afinal XOR-gate 560 is applied to the final two intermediate signals ofthe previous layer. The output of the last XOR gate 560 is the feedbacksignal output by the adder 540 and the WX-gate 500. It is provided tothe input of the corresponding register 210 as shown in FIG. 7.

Naturally, also in terms of the adder 540 a different configuration fromthe previously described employing, for instance, XNOR-gates may beused. Moreover, also a different arrangement, eventually requiring morethan the 31 XOR-gates or XNOR-gates may be implemented.

Moreover, in other embodiments according to the present invention, adifferent number of registers 210 than 32 registers and relatedcircuitry may be implemented. In the case of a cascaded adder like theadder 540, it may be possible to implement a similar adder with L layersfor a state machine 200 comprising more than 2^(L−1) but not more than2^(L) registers. Since every XOR-gate or XNOR-gate comprises two inputsand one output and reduces therefore the number of signals by one, asimilar implementation of a WX-gate 500 may be realized by using thesame number of masking gates as the number of registers and one XOR-gateor one XNOR-gate less than the number of registers.

When, for instance, the possibility of different generator polynomialsor, in more general terms, the number of different feedback functions islimited such that some elements of the vectors a_(j) or some matrixelements A_(ij) are constant (e.g. equal to 0) for the differentfeedback functions, the corresponding AND-gates 550 along with thecorresponding XOR-gates 560 may eventually be dropped. Thereby, asimplification of the circuitry may be achieved.

For the sake of completeness, it should be noted that the state machineas illustrated in FIGS. 6 and 7, according to the embodiment of thepresent invention, may furthermore comprise one or more memory elementssuch as flip-flops, registers, SRAM-cells or registers to store theelements of the multi-transition matrix A 340. In other words, the statemachine 200 as shown in FIGS. 6 and 7 may, for instance, comprise aregister to store 32·32=1024 (binary) matrix elements of the matrix A340 as registered values (in the embodiment described above).

To illustrate that the state machine 200 according to an embodiment ofthe present invention along with the generator 300 according to anembodiment of the present invention operating on the previouslypresented matrix-related description, do in fact yield the same result,FIG. 8 shows a direct comparison of an output of a linear feedback shiftregister 100 as shown in FIG. 1 a and that of a corresponding statemachine 200 as described in the context of FIGS. 6 and 7.

FIG. 8 a illustrates the previously described single transition matrix T330 for a linear feedback shift register 100 with N=4 registers 110. Asdescribed in the context of FIGS. 4 and 5, the single transition matrixT comprises in the first row, the generator polynomial coefficients 310of the corresponding polynomial 400 g(x) according to equation (1).

However, different from the matrix 410 of FIG. 4, the single transitionmatrix 330 as shown in FIG. 8 a is based on an 8×8 matrix for G=8entries. As a consequence, the signal transition matrix T 330 comprises,apart from the generator polynomial coefficients 310, also the columnvector 440 comprising zeros and the shift matrix S 450 as outlined incontext of FIG. 5.

FIG. 8 b shows a sequence of 18 bits output by the corresponding LFSR100 as shown in FIG. 1 a at the output of register 110-4 which isindicative of the state x₄ of this register.

FIG. 8 c shows the resulting multi-transition matrix A 340 obtained by agenerator 300 according to an embodiment of the present invention basedon the generator polynomial coefficients 310 and the resultingintermediate single transition matrix T 330 as shown in FIG. 8 a for R=6bits to be provided in the framework of a pseudo-random word of thepseudo-random word stream generated by a state machine 200 according toan embodiment of the present invention. In other words, FIG. 8 c showsthe multi-transition matrix A=T⁶ based on the single transition matrix Tshown in FIG. 8 a.

Starting from the same initial states of the registers 210 compared tothe states of the corresponding registers 110 of the linear feedbackshift register 100, the state machine 200 as shown in FIGS. 6 and 7provided with the multi-transition matrix A as shown in FIG. 8 cprovides the word stream shown in FIG. 8 d comprising the same 18pseudo-random bits as shown in FIG. 8 b. In other words, both sequencesare identical, which proves the concept.

Apart from the previously described matrix-implementation of embodimentsaccording to the present invention in the form of a state machine 200and in the form of a generator 300 also further straight-forwardimplementations may be realized, which are, however, at leasttheoretically based on the same matrix description. Returning to FIG. 2,the feedback function of the feedback circuit 230 may also beprogrammable by providing the feedback circuit 230 via the input 260with a truth table to define each of the feedback signals on the basisof the states of the registers 210. Such a truth table-likeimplementation may, for instance be realized by employing a memory-likestructure in which all possible different states of the state machine200 may be considered as an address input signal, wherein thecorresponding values stored in the memory corresponds the generatedfeedback signal. Such an implementation may be realized on the basis ofRAM techniques (RAM=Random Access Memory), NVM techniques(NVM=Non-Volatile Memory) or ROM techniques (ROM=Read-Only Memory),depending on whether the feedback function is supposed to beprogrammable (RAM, NVM implementation) and whether the truth table issupposed to be stored in a non-volatile way (NVM, ROM implementation).Naturally, in the case of a non-programmable implementation based on aROM technology, an optimization may, for instance, be employed by onlyimplementing memory cells for either feedback signals which are equal toone or equal to zero, depending on further details of the embodiment.

An alternative truth table-based implementation may be that of a FPGA(FPGA=Field Programmable Gate Array), in which a logical expressiondescribing the behavior of the corresponding state machine 200 accordingto an embodiment of the present invention may be stored utilizinglook-up-tables and other elements of typical FPGA implementation.

Naturally, in all of the previously described straight-forwardimplementations, an optimization based on a logical description of thecorresponding feedback function using the rules of Boolean algebra maybe utilized. By starting from a truth table and by isolating allcombinations of states which lead to a feedback signal having the valueof one and by concatenating the corresponding logic expressions (e.g.based on a plurality of AND-combinations) may be a starting point forsuch an optimization. The optimization may then use, for instance, thedifferent rules of Boolean algebra to transform the resulting expressioninto a more concise expression which may then, for instance, beimplemented using the previously described FPGA structure.

Furthermore, it should be noted that the different embodiments accordingto the present invention may for instance be combined in the frameworkof a system. In other words, a generator 300 according to an embodimentof the present invention as shown in FIG. 3 may be coupled via itsoutput 350 to the input 260 of the feedback circuit 230 of a statemachine 200 according to an embodiment of the present invention. As aconsequence, this may lead to a very concise and flexible system beingcapable of providing a pseudo-random word stream or pseudo-random bitstream based on a generation polynomial coefficient and the number ofbits per word.

Moreover, it should be noted that the systems, state machines andgenerators according to embodiments of the present invention describedabove also correspond to a description of respective methods accordingto the present invention. In other words, to some extent the figuresdescribed before also reflect flow charts of corresponding methodsaccording to embodiments of the present invention.

Depending on certain implementation requirements of embodimentsaccording to the present invention, embodiments of inventive methods canbe implemented in hardware or in software. The implementation can beperformed using a digital storage medium, in particular, a disc, a CD ora DVD having electronically readable control signals stored thereon,which cooperate with a programmable computer or processor such that anembodiment of the inventive method is performed. Generally, anembodiment of the present invention is, therefore, a computer programproduct where the program code is stored on a machine-readable carrier,the program code being operative for performing an embodiment of theinventive methods, when the computer program runs on the computer orprocessor. In other words, embodiments of the inventive method are,therefore, a computer program having a program code for performing atleast one of the embodiments of the inventive methods, when the computerprogram runs on the computer or processor. A processor can be formed bycomputer, a chip card, a smart card, an application-specific integratedcircuit (ASIC), a system-on-chip (SOC) or another integrated circuit(IC).

While this invention has been described in terms of several embodiments,there are alterations, permutations, and equivalents which fall withinthe scope of this invention. It should also be noted that there are manyalternative ways of implementing the methods and compositions of thepresent invention. It is therefore intended that the following appendedclaims be interpreted as including all such alterations, permutationsand equivalents as fall within the true spirit and scope of the presentinvention.

The invention claimed is:
 1. A state machine for generating apseudo-random word stream, each word of the word stream comprising aplurality of subsequent bits of a pseudorandom bit sequence, the statemachine comprising: a plurality of clocked registers, the registerscomprising corresponding inputs and corresponding outputs; and afeedback circuit coupled to the inputs and to the outputs of theregisters and adapted to provide a plurality of feedback signals to theinputs of the registers based on a feedback function and the pluralityof register output signals of the registers, the register output signalsbeing indicative of states of the registers, wherein the state machineis configured such that a first word defined by the plurality ofregister output signals comprises a first set of subsequent bits of thepseudo-random bit stream and such that a subsequent second word definedby the plurality of register output signals comprises a second set ofsubsequent bits of the pseudo-random bit stream; wherein the feedbackcircuit comprises a logic stage and an adder; wherein the logic stage isconfigured to provide a plurality of logic stage output signals on thebasis of the output signals of the registers by selectively weighing theoutput signals of the registers depending on a set of multi-transmissionmatrix elements; wherein the logic stage comprises a masking gateconfigured to provide a logic stage output signal depending on an outputsignal of one of the registers and one of the multi-transition matrixelements; wherein the adder is configured to provide at least onefeedback signal using a modulo-2 summation of the logic stage outputsignals; and wherein the adder comprises a plurality of XOR-gates orXNOR-gates for the modulo-2 summation.
 2. The state machine according toclaim 1, wherein the feedback circuit is adapted such that the pluralityof feedback signals provided to the inputs of the registers are based ona single application of the feedback function leading to new states ofthe registers, the new states representing a state of a linear feedbackshift register initialized with the original states of the registersafter being clocked multiple times.
 3. The state machine according toclaim 1, wherein the feedback circuit is configured to provide at leasttwo feedback signals based on register output signals of at least twodifferent resisters.
 4. The state machine according to claim 1, whereinthe feedback circuit is configured such that the feedback signals dependonly on the feedback function and the plurality of register outputsignals.
 5. The state machine according to claim 1, wherein the feedbackcircuit is configured such that the second word immediately follows thefirst word in response to a clock signal.
 6. The state machine accordingto claim 1, wherein the state machine is adapted to provide thepseudo-random bit sequence with a frequency being higher than afrequency of the clock signal provided to the registers by a factorlarger than
 1. 7. The state machine according to claim 1, wherein thefeedback circuit is configured so that the feedback function isprogrammable.
 8. The state machine according to claim 1, wherein thefeedback circuit is configured such that at least one feedback signal ofthe plurality of feedback signals is provided based on a modular-2scalar product of a set of multi-transmission matrix elements and valuesof the register output signals.
 9. The state machine according to claim1, wherein the adder comprises a cascade of layers of XOR-gates orXNOR-gates to reduce a number of signals in each layer in the cascade oflayers by approximately a factor of
 2. 10. The state machine accordingto claim 9, wherein the cascade of layers of the adder comprises Llayers, wherein the plurality of clocked registers comprises more than2^(L−1) and less than or equal to 2^(L) registers, and wherein L is aninteger.
 11. A generator for generating a description of a state machinefeedback function on the basis of a set of generator polynomialcoefficients defining configuration of a linear feedback shift register,the generator comprising: a calculator configured to calculate thedescription of the state machine feedback function for a state machineon the basis of the set of generator polynomial coefficients, whereinthe calculator is configured to determine the feedback function suchthat a state transition between immediately subsequent states of thestate machine defined by the feedback function corresponds to a sequenceof state transitions of a linear feedback shift register, the linearfeedback shift register being configured in accordance with thegenerator polynomial coefficients; wherein the calculator is configuredto acquire, as an intermediate result, a single transition matrixdescription of a state machine feedback function on the basis of thegenerator polynomial coefficients of the linear feedback shift register,the single transition matrix description describing a relationshipbetween an initial state of a state machine and a consecutive state ofthe state machine according to the generator polynomial coefficients fora single linear feedback shift register state transition; wherein thecalculator is configured to compute a power of the single transitionmatrix description to acquire, as the state machine feedback function, amulti-transition matrix description describing, in a combined form, aplurality of linear feedback shift register state transitions; andwherein the calculator is adapted to calculate the multi-transitionmatrix description A comprising the multi-transition matrix elementsA_(ij) based on the equationA=T ^(R) mod 2 wherein T is the single transition matrix descriptioncomprising the single transition matrix elements T_(ij) wherein i is arow index in the range between 1 and G, wherein j is a column index inthe range between 1 and G, wherein N is a number of registers of thelinear feedback shift register, wherein R is an integer larger than N,wherein G is an integer larger than or equal to a maximum of N and R,wherein T_(1j) for j in the range between 1 and N is equal to the set ofgenerator polynomial coefficients, wherein T_(ij) for i=j+1 and j in therange between 1 and (G−1) is equal to 1, and wherein T_(ij) is equal to0.
 12. The generator according to claim 11, wherein the calculator isconfigured such that a single state transition of the state machinebased on the determined feedback function corresponds to the sequence ofstate transitions of the linear feedback shift register.
 13. Thegenerator according to claim 11, wherein the calculator is configuredsuch that the state transitions of the state machine starting frominitial states of the registers correspond to states of the linearfeedback shift register starting from the same states after a sequenceof state transitions of the linear feedback shift register.
 14. Thegenerator according to claim 11, wherein the generator is adapted suchthat the description of the state machine feedback function comprisesonly binary units.
 15. The generator according to claim 11, wherein thegenerator adapted such that the set of generator polynomial coefficients(q_(N), q_(N−1), . . . , q₁) corresponds to A polynomial${g(x)} = {1 + {\sum\limits_{n = 1}^{N}\;{q_{n}x^{n}}}}$ of the linearfeedback shift register comprising N registers wherein q_(n) is 0 or 1for n=1, . . . , N.
 16. The generator according to claim 11, wherein thegenerator is adapted such that the set of generator polynomialcoefficients comprises at least two non-vanishing generator polynomialcoefficients.
 17. A method for generating a description of a statemachine feedback function on the basis of a set of generator polynomialcoefficients defining a configuration of a linear feedback shiftregister, the method comprising: calculating the description of thestate machine feedback function for a state machine on the basis of theset of generator polynomial coefficients such that a state transitionbetween immediately subsequent states of the state machine defined bythe feedback function corresponds to a sequence of state transitions ofthe linear feedback shift register configured in accordance with agenerator polynomial coefficients wherein calculating the descriptioncomprises generating a set of parameter coefficients A_(ij) such thatA=T ^(R) mod 2 wherein A_(ij) are elements of the matrix A, wherein T isa matrix comprising elements T_(ij), wherein i is a row index in therange between 1 and G, wherein j is a column index in the range between1 and G, wherein G is an integer larger than or equal to a maximum of Nand R, wherein A_(1j) for j in the range between 1 and N is given by theset of generator polynomial coefficients, wherein T_(ij) for i=j+1 and jin the range between 1 and (G−1) is equal to 1, wherein T_(ij) is elseequal to
 0. 18. A method for generating a pseudo-random word streamusing a state machine, each word z(k) of the word stream comprising Rbits, based on a set of parameter coefficients A_(ij) according toA=T ^(R) mod 2 wherein A_(ij) are elements of a matrix A, wherein T is amatrix comprising elements Tij 1 wherein i is a row index in the rangebetween 1 and G, wherein j is a column index in the range between 1 andG, wherein G is an integer larger than or equal to a maximum of N and R,wherein A_(1j) for j in the range between 1 and N is given by a set ofgenerator polynomial coefficients comprising N bits indicative of afeedback polynomial, wherein N is an integer larger than 1, wherein R isa positive integer larger than N; wherein T_(ij) for and i=j+1 in therange between 1 and (G−1) is equal to 1, wherein T_(ij) is else equal to0, the method comprising: generating the word z(k) in a plurality ofclocked registers by providing feedback signals from a feedback circuitto inputs of the clocked registers such thatz(k+1)=A·z(k)mod 2 wherein z_(j)(k) is a vector element of the word z(k)in the form of a vector, wherein k is a time index.
 19. A non-transitorycomputer readable medium, that store instructions for executing aprogram, when executed by a processor, that performs the methodaccording to claim
 18. 20. A state machine for generating apseudo-random word stream, each word of the word stream comprising aplurality of subsequent bits of a pseudo-random bit sequence, the statemachine comprising: a plurality of clocked registers, the registerscomprising corresponding inputs and corresponding outputs; and afeedback circuit coupled to the inputs and to the outputs of theregisters and adapted to provide a plurality of feedback signals to theinputs of the registers based on a feedback function and the pluralityof register output signals of the registers, the register output signalsbeing indicative of states of the registers, wherein the state machineis configured such that a first word defined by the plurality ofregister output signals comprises a first set of subsequent bits of thepseudo-random bit stream and such that a subsequent second word definedby the plurality of register output signals comprises a second set ofsubsequent bits of the pseudo-random bit stream; wherein the feedbackcircuit is adapted such that the plurality of feedback signals providedto the inputs of the registers are based on a single application of thefeedback function leading to new states of the registers, the new statesrepresenting a state of a linear feedback shift register initializedwith the original states of the registers after being clocked multipletimes; and wherein the feedback function comprises a set of parametercoefficients A_(ij) such thatA=T ^(R) mod 2 wherein A_(ij) are elements of the matrix A, wherein is amatrix comprising elements T_(ij), wherein i is a row index in the rangebetween 1 and G, wherein j is a column index in the range between 1 andG, wherein G is an integer larger than or equal to a maximum of N and R,wherein A_(1J) for j in the range between 1 and N is given by the set ofgenerator polynomial coefficients, wherein T_(ij) for i=j+1 and j in therange between 1 and (G−1) is equal to 1, wherein T_(ij) is else equal to0.
 21. A system for generating a pseudo-random word stream, each word ofthe word stream comprising a plurality of subsequent bits of apseudo-random bit sequence, the system comprising: a state machineaccording to any of the claims 1 and 20; and a generator for generatinga description of a state machine feedback function on the basis of a setof generator polynomial coefficients defining a configuration of alinear feedback shift register, the generator comprising: a calculatorconfigured to calculate the description of the state machine feedbackfunction for a state machine on the basis of the set of generatorpolynomial coefficients, wherein die calculator is configured todetermine the feedback function such that a state transition betweenimmediately subsequent states of the state machine defined by thefeedback function corresponds to a sequence of state transitions of alinear feedback shift register, the linear feedback shift register beingconfigured in accordance with the generator polynomial coefficients;wherein the calculator is configured to acquire, as an intermediateresult, a single transition matrix description of a state machinefeedback function on the basis of the generator polynomial coefficientsof the linear feedback shift register, the single transition matrixdescription describing a relationship between an initial state of astate machine and a consecutive state of the state machine according tothe generator polynomial coefficients for a single linear feedback shiftregister state transition; wherein the calculator is configured tocompute a power of the single transition matrix description to acquire,as the state machine feedback function, a multi-transition matrixdescription describing, in a combined form, a plurality of linearfeedback shift register state transitions; and wherein the calculator isadapted to calculate the multi-transition matrix description Acomprising the multi-transition matrix elements A_(ij) based on theequationA=T ^(R) mod 2 wherein T is the single transition matrix descriptioncomprising the single transition matrix elements T_(ij), wherein i is arow index in the range between 1 and G, wherein j is a column index inthe range between 1 and G, wherein N is a number of registers of thelinear feedback shift register, wherein R is an integer larger than N,wherein G is an integer larger than or equal to a maximum of N and R,wherein T_(1j) for j in the range between 1 and N is equal to the set ofgenerator polynomial coefficients, wherein T_(ij) for i=j+1 and j in therange between 1 and (G−1) is equal to 1, and wherein T_(ij) is equal to0, wherein the feedback circuit of the state machine is adapted suchthat the feedback signals are provided based on the description of thestate machine feedback function calculated by the generator.
 22. Thesystem according to claim 21, wherein the system is adapted to provideat least two feedback signals based on at least two different states ofthe registers of the state machine.
 23. A system for generating apseudo-random word stream, each word of the word stream comprising aplurality of subsequent bits of a pseudo-random bit sequence, the systemcomprising: a state machine including, a plurality of clocked registers,the registers comprising, corresponding inputs and correspondingoutputs; a feedback circuit coupled to the inputs and to the outputs ofthe registers and adapted to provide a plurality of feedback signals tothe inputs of the registers based on a feedback function and theplurality of register output signals of the registers, the registeroutput signals being indicative of states of the registers, wherein thestate machine is configured such that a first word defined by theplurality of register output signals comprises a first set of subsequentbits of the pseudo-random bit stream and such that a subsequent secondword defined by the plurality of register output signals comprises asecond set of subsequent bits of the pseudo-random bit stream; and acircuit to provide a word of the pseudo-random bit stream in asequentially encoded form; and a generator for generating a descriptionof a state machine feedback function on the basis of a set of generatorpolynomial coefficients defining a configuration of a linear feedbackshift register, the generator including: a calculator configured tocalculate the description of the state machine feedback function for astate machine on the basis of the set of generator polynomialcoefficients, wherein the calculator is configured to determine thefeedback function such that a state transition between immediatelysubsequent states of the state machine defined by the feedback functioncorresponds to a sequence of state transitions of a linear feedbackshift register, the linear feedback, shift register being configured inaccordance with the generator polynomial coefficients; wherein thecalculator is configured to acquire, as an intermediate result, a singletransition matrix description of a state machine feedback function onthe basis of the generator polynomial coefficients of the linearfeedback shift register, the single transition matrix descriptiondescribing a relationship between an initial state of a state machineand a consecutive state of the state machine according to the generatorpolynomial coefficients for a single linear feedback shift registerstate transition; wherein the calculator is configured to compute apower of the single transition matrix description to acquire, as thestate machine feedback function, a multi-transition matrix descriptiondescribing, in a combined form, a plurality of linear feedback shiftregister state transitions; and wherein the calculator is adapted tocalculate the multi-transition matrix description A comprising themulti-transition matrix elements A_(ij) based on the equationA=T ^(R) mod 2 wherein T is the single transition matrix descriptioncomprising the single transition matrix elements T_(ij), wherein i is arow index in the range between 1 and G, wherein j is a column index inthe range between 1 and G, wherein N is a number of registers of thelinear feedback shift register, wherein R is an integer larger than N,wherein G is an integer larger than or equal to a maximum of N and R,wherein T_(ij) for j in the range between 1 and N is equal to the set ofgenerator polynomial coefficients, wherein T_(ij) for i=j+1 and j in therange between 1 and (G−1) is equal to 1, and wherein T_(ij) is equal to0, wherein the feedback circuit of the state machine is adapted suchthat the feedback signals are provided based on the description of thestate machine feedback function calculated by the generator.